Zeta function (operator)
The zeta function of a mathematical operator [math]\displaystyle{ \mathcal O }[/math] is a function defined as
- [math]\displaystyle{ \zeta_{\mathcal O}(s) = \operatorname{tr} \; \mathcal O^{-s} }[/math]
for those values of s where this expression exists, and as an analytic continuation of this function for other values of s. Here "tr" denotes a functional trace.
The zeta function may also be expressible as a spectral zeta function[1] in terms of the eigenvalues [math]\displaystyle{ \lambda_i }[/math] of the operator [math]\displaystyle{ \mathcal O }[/math] by
- [math]\displaystyle{ \zeta_{\mathcal O}(s) = \sum_{i} \lambda_i^{-s} }[/math].
It is used in giving a rigorous definition to the functional determinant of an operator, which is given by
- [math]\displaystyle{ \det \mathcal O := e^{-\zeta'_{\mathcal O}(0)} \;. }[/math]
The Minakshisundaram–Pleijel zeta function is an example, when the operator is the Laplacian of a compact Riemannian manifold.
One of the most important motivations for Arakelov theory is the zeta functions for operators with the method of heat kernels generalized algebro-geometrically.[2]
See also
References
- ↑ Lapidus & van Frankenhuijsen (2006) p.23
- ↑ Soulé, C.; with the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer (1992), Lectures on Arakelov geometry, Cambridge Studies in Advanced Mathematics, 33, Cambridge: Cambridge University Press, pp. viii+177, ISBN 0-521-41669-8
- Lapidus, Michel L.; van Frankenhuijsen, Machiel (2006), Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings, Springer Monographs in Mathematics, New York, NY: Springer-Verlag, ISBN 0-387-33285-5
- Fursaev, Dmitri; Vassilevich, Dmitri (2011), Operators, Geometry and Quanta: Methods of Spectral Geometry in Quantum Field Theory, Theoretical and Mathematical Physics, Springer-Verlag, p. 98, ISBN 978-94-007-0204-2
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